B Thought Experiment from a Non-Physicist

[StoneWaz]

Hello All. This is driving me crazy and nobody can give me a straight answer, if there is one. I was daydreaming in my yard last week and came up with a weird question. Let's say I discovered a hole in my yard. The hole is plenty wide and well-constructed and has a series of rungs I can easily climb down. This hole actually goes straight down, through the exact center of the Earth, and pops out the other side. I climb down the rungs into the hole. Now let's ignore all factors such as heat/pressure/breathable air/etc. and let's focus on gravity. At some point the direction of gravity would have to shift, right? And that would occur at the exact mid-point, right? But does gravity increase or decrease as I go further down and what, exactly, happens at the center -- do I find myself suddenly flipped around and climbing the rungs or would there be a point where gravity is "neutral" and I would be floating? Or would the "epi-center" of the gravity shift be so tiny as to render such a question meaningless?
Please help my twisted mind resolve this. Thanks!

 

[Mark44]

At some point the direction of gravity would have to shift, right? And that would occur at the exact mid-point, right?

Yes and yes. At the top of the whole, all of the Earth's mass is pulling on you (and you on it). As you get farther down in the hole, there is less mass below you, and more above you. At the exact center, all gravitation forces would essentially cancel.

But does gravity increase or decrease as I go further down and what, exactly, happens at the center

Decreases to zero, and then begins increasing again. When you pop out the other side, gravity will be the same as at the "top" of the hole.

 

[Ibix]

Mark44's answer is correct assuming an idealised Earth that is the same density all the way down. The actual Earth is denser at the core, and in fact gravity increases as you go down because you are getting closer to the dense core, until you get close to the core. Then the effect Mark44 is talking about starts to dominate and gravity drops to zero at the centre.

 

[StoneWaz]

Amazing stuff to think about. Thanks for the help!

 

[Janus]

Here's a graph comparing how gravity would change with depth according to three different models for its interior: constant density, Density that increases linearly as you move towards the center, and Our estimate of what is is for the real Earth (Preliminary Reference Earth Model,or PREM)

 

[Chandra Prayaga]

Here is something else that is interesting. In the idealized Earth mentioned in post #3, you don't need rungs to climb down and then up on the other side. If you let go of the rungs, you will oscillate back and forth from one end of the tunnel to the other, and back again, like a ball attached to a spring.

 

[Ibix]

In the idealized Earth mentioned in post #3, you don't need rungs to climb down and then up on the other side. If you let go of the rungs, you will oscillate back and forth from one end of the tunnel to the other, and back again, like a ball attached to a spring.

That would work in any spherically symmetric Earth regardless of the radial density distribution, in fact. Gravitational potential is equal at all points on the surface, so conservation of energy means you come to a stop at the same radius you started at (neglecting friction). So it would probably work in more general cases than the spherically symmetric Earth, too.

 

[jbriggs444]

Here is something else that is interesting. In the idealized Earth mentioned in post #3, you don't need rungs to climb down and then up on the other side. If you let go of the rungs, you will oscillate back and forth from one end of the tunnel to the other, and back again, like a ball attached to a spring.

Works best if the hole is evacuated and runs from north pole to south pole, of course.

 

[StoneWaz]

And would these forces be strong enough to kill me -- i.e., could a human make it through the center? Ignoring the other variables, of course...

 

[Janus]

And would these forces be strong enough to kill me -- i.e., could a human make it through the center? Ignoring the other variables, of course...

You'd be in free fall the whole way and the tidal gradient wouldn't be anything to worry about. As long as you don't hit a wall on the way through (at the center you'd be moving at better than 7.9 km/sec), you'd be fine.

 

[jbriggs444]

And would these forces be strong enough to kill me -- i.e., could a human make it through the center? Ignoring the other variables, of course...

As long as the walls of the tube are strong enough to withstand the pressure (i.e. they are made of unobtainium), as long as the tube is evacuated and as long as one ignores Coriolis, falling through the tube would be like being weightless. There would be no stresses worth worrying about.

Edit: Nosed out by @Janus

 

[Janus]

Some additional points that might be of interest to the OP.
If we we assume a spherical Earth of constant density, and using a radius of 6378 km for the Earth, then the round trip would take 84 min 28 sec, or 42 min 14 sec one way.

If we drill our hole such that it doesn't pass through the center, but at some other angle so that it comes out at a point other than the exact opposite point of the Earth, an object dropped down that hole (we will have to assume friction-less sides here) will also take 42 min 14 sec to reach the other end.
So for instance, if you were to drill a straight hole through the Earth from LA to New York, put a ball in it and let it "roll" from one end to the other (again ignoring friction, the Earth's rotation, etc. ), it would take 42 min 14 sec to travel from LA to New York. This is the idea behind the theoretical concept known as a gravity train:
https://en.wikipedia.org/wiki/Gravity_train

The time for the round trip is equal to the time it would take for a satellite to complete one orbit of the Earth at its surface.

 

[StoneWaz]

Wow, again. A lot to be amazed by. So, in summary: I would step into the hole, I would drop down the tunnel (very quickly) and I would have just enough "force" on to pop out of the hole on the other side (in 42 minutes and 14 sec), then gravity would take me back "down" and the cycle would repeat over and over. And if had not changed position in transit, I emerge feet up on the other side?!
Thanks again for the enlightenment. I love this stuff!

 

[pikpobedy]

Rolling involves energy being stored as rotational KE in addition to translation KE.